A MORE COMPLETE THERMODYNAMIC FRAMEWORK FOR SOLID CONTINUA

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Issue Date

2015-05

Author

Surana, Karan S.

Powell, Michael J.

Reddy, J. N.

Publisher

Yildiz Technical University

Type

Article

Article Version

Scholarly/refereed, publisher version

Rights

Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.

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Abstract

The Jacobian of deformation at a material point can be decomposed
into the stretch tensor and the rotation tensor. Thus, varying
Jacobians of deformation at the neighboring material points
in the deforming volume of solid continua would yield varying
stretch and rotation tensors at the material points. Measures of
strain, such as Green’s strain, at a material point are purely a
function of the stretch tensor, i.e. the rotation tensor plays no
role in these measures. Alternatively, we could also consider
decomposition of displacement gradient tensor into symmetric
and skew symmetric tensors. Skew symmetric tensor is also a
measure of pure rotations whereas symmetric tensor is a measure
of strains, i.e. stretches. The measures of rotations in these
two approaches describe the same physics but are in different
forms. Polar decomposition gives the rotation matrix and not
the rotation angles whereas the skew symmetric part of the displacement
gradient tensor yields rotation angles that are explicitly
and conveniently defined in terms of the displacement gradients.
The varying rotations and rotation rates arise in all deforming
solid continua due to varying deformation of the continua at
neighboring material points, hence are internal to the volume
of solid continua and are explicitly defined by the deformation,
therefore do not require additional degrees of freedom to define
them. If the internal varying rotations and their rates are
resisted by the continua, then there must exist internal moments
corresponding to these. The internal rotations and their rates and
the corresponding moments can result in additional energy storage
and dissipation. This physics is all internal to the deforming
continua (hence does not require consideration of additional
external degrees of freedom and associated external moments)
and is neglected in the presently used continuum theories for
isotropic, homogeneous solid continua. The continuum theory
presented in this paper considers internal varying rotations and
associated conjugate moments in the derivation of the conservation
and balance laws, thus the theory presented in this paper is
“a polar theory for solid continua” but is different than the micropolar
theories published currently in which material points
have six external degrees of freedom i.e. rotations are additional
external degrees of freedom.
This polar continuum theory only accounts for internal rotations
and associated moments that exist as a consequence of
deformation but are neglected in the present theories. We call
this theory “a polar continuum theory” as it considers rotations
and moments as conjugate pairs in a deforming solid continua
though these are internal, hence are purely related to the deformation
of the solid. It is shown that the polar continuum theory
presented in this paper is not the same as the strain gradient
theories reported in the literature. The differences are obviously
in terms of the physics described by them and the mathematical
details associated with conservation and balance laws. In
this paper, we only consider polar continuum theory for small
deformation and small strain. This polar continuum theory presented
here is a more complete thermodynamic framework as
it accounts for additional physics of internally varying rotations
that is neglected in the currently used thermodynamic framework.
This thermodynamic framework is suitable for isotropic,
homogeneous solid matter such as thermoelastic and thermoviscoelastic
solid continua with and without memory when the
deformation is small. The paper also presents preliminary material
helpful in consideration of the constitutive theories for polar
continua.

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Except where otherwise noted, this item's license is described as: Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.